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Find the sum of the series .^(n)C(0)+2...

Find the sum of the series
`.^(n)C_(0)+2.^(n)C_(1)x+3.^(n)C_(2)x^(2)+....+(n+1).^(n)C_(n)x^(n)`
and hence show that ,
`.^(n)C_(0)+2.^(n)C_(1)x+3.^(n)C_(2)x^(2)+....+(n+1)^(n)C_(n)=(n+2)2^(n-1)`

Text Solution

Verified by Experts

The correct Answer is:
`(1+x)^(n-1)[(n+1)x+1]`
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CHHAYA PUBLICATION-BINOMIAL THEOREM -EXERCISE ( Long answer Type Questions )
  1. If (1+x)^(n)=C(0)+C(1)x+C(2)x^(2)+.....+C(n)x^(n) then show :

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  2. If (1+x)^(n)=C(0)+C(1)x+C(2)x^(2)+.....+C(n)x^(n) then show :

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  3. If (1+x)^(n)=C(0)+C(1)x+C(2)x^(2)+.....+C(n)x^(n) then show :

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  4. If (1+x)^(n)=C(0)+C(1)x+C(2)x^(2)+.....+C(n)x^(n) then show : 3.^(...

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  5. The power of a of 10thterm in the extension (a+2b)^20-

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  6. (1-x)^(n)=C(0)-C(1)x+C(2)x^(2)-C(3)x^(3)+...+C(r)(-1)^(r)x^(r)+.....+(...

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  7. Find the sum of the series .^(n)C(0)+2.^(n)C(1)x+3.^(n)C(2)x^(2)+......

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  8. Show : x-^(n)C(1)(x+y)+^(n)C(2)(x+2y)-^(n)C(3)(x+3y)+....=0

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  9. Show : 2^(n)-(n)/(1!).2^(n-1)+(n(n-1))/(2!).2^(n-2)-....+(-1)^(n)=1

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  10. If (1+x)^(n)=C(0)+C(1)x+C(2)x^(2)+...+C(n)x^(n) "show that ", (1)/(1...

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  11. Find numerically the greatest terms in the following expansion : (1...

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  12. Find numerically the greatest terms in the following expansion : (1-...

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  13. Find numerically the greatest terms in the following expansion : (2...

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  14. Find numerically the greatest terms in the following expansion : [1...

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  15. Find numerically the greatest terms in the following expansion : {(2x...

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  16. Find numerically the greatest terms in the following expansion (ax-...

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  17. Find numerically the greatest cofficients in the following expansions ...

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  18. Find numerically the greatest cofficients in the following expansions ...

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  19. Which terms in the expansion of (x+(1)/(2x))^(3n) has the greatest coe...

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  20. In the expansion of the expression (a+x)^(15) ,if the eleventh term i...

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